Learning ultra-compressible hyperelasticity with splines: Constitutive asymmetries and non-unique representations

2026-04-15

AI Summary

This study addresses the limitations of traditional hyperelastic models in consistently capturing the asymmetric large-deformation behavior of highly compressible foam materials under tension and compression, as well as the invalidity of the implicit assumption of uniqueness in strain energy functions. To overcome these issues, the authors propose a spline-based data-driven framework that constructs both separable and non-separable forms of the strain energy density function in the invariant space of $(\bar{I}_1, \bar{I}_2, J)$, effectively coupling isochoric and volumetric deformation responses. By fitting multi-axial experimental data, the model successfully reproduces the complex mechanical behavior of ultralight foams under tension, compression, and shear. The work systematically reveals, for the first time, the inherent non-uniqueness of energy representations in hyperelastic modeling and demonstrates both the limitations of conventional low-parameter models and the necessity of incorporating coupling terms.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Large Multimodal Models (LMMs)

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Highly compressible solids, such as foams, exhibit complex responses, including pronounced tension-compression asymmetry. Capturing such behaviors within unified hyperelastic frameworks remains challenging. Invariant-based hyperelastic models are commonly identified from standard tests such as homogeneous uniaxial tension/compression and simple shear, implicitly assuming a unique energy representation. Here we show that this assumption is fundamentally violated and that, oftentimes, the choice of which term should prevail is just a matter of taste. Using spline-based strain-energy density functions as a data-adaptive tool and stress-strain experimental data for elastomeric foams, we expose this non-uniqueness, often hidden in low-parameter formulations. Our framework captures the volumetric deformation of ultra-light foams used in racing shoes using homogeneous experimental data from tension, compression, and shear. We formulate an overly rich ansatz of separable and non-separable energies in the ($\bar{I}_1$, $\bar{I}_2$, $J$) space à la Money-Rivlin. These constructs, defined by multiplicative decompositions, resemble classical invariant-based models while generalizing them to a data-driven spline representation. This serves two purposes: (i) to capture the response under complex volumetric deformation modes and (ii) to allow non-uniqueness in the identification problem to emerge naturally. We find that a coupling term between isochoric and volumetric deformation, such as $Ψ(\bar{I}_1,J)$ or $Ψ(\bar{I}_2,J)$, is essential and that additional coupling terms help but are not fully necessary; rather, they pronounce the non-uniqueness. As a consequence, different models may be indistinguishable on available data. Importantly, these challenges are not specific to splines but extend to traditional and neural network-based models.

Problem

• hyperelasticity
• tension-compression asymmetry
• constitutive modeling
• non-uniqueness
• ultra-compressible solids

Innovation

• spline-based hyperelasticity
• tension-compression asymmetry
• non-unique constitutive representation
• volumetric-isochoric coupling
• data-driven constitutive modeling